Consider the following situation. Emma (the agent) works for Rachel (the principal). Emma's work has three possible outcomes (that is, profit to Rachel): Profits e=0 e=1 $0 p=0.6 p=0.1 p=0.3 p=0.3 $2,000 $3,000 p=0.1 p=0.6 where e=0 means that Emma puts no effort on her job, and e = 1 means that Emma puts effort on her job. Each cell shows the associated probabilities on each outcome. Emma's utility function is given by
Consider the following situation. Emma (the agent) works for Rachel (the principal). Emma's work has three possible outcomes (that is, profit to Rachel): Profits e=0 e=1 $0 p=0.6 p=0.1 p=0.3 p=0.3 $2,000 $3,000 p=0.1 p=0.6 where e=0 means that Emma puts no effort on her job, and e = 1 means that Emma puts effort on her job. Each cell shows the associated probabilities on each outcome. Emma's utility function is given by
Chapter8: Game Theory
Section: Chapter Questions
Problem 8.12P
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Transcribed Image Text:Consider the following situation. Emma (the agent) works for Rachel (the principal). Emma's
work has three possible outcomes (that is, profit to Rachel):
Profits e=0
$0
p=0.6
$2,000
p = 0.3
$3,000 p=0.1
U=
e = 1
p=0.1
p = 0.3
p=0.6
where e = 0 means that Emma puts no effort on her job, and e = 1 means that Emma puts effort
on her job. Each cell shows the associated probabilities on each outcome.
Emma's utility function is given by
√ if she puts no effort,
√w-5 if she puts effort,
where w the payment from Rachel. If Emn does not work for Rachel, Emma will have an
outside option that gives her the utility level of 15.
(a) Calculate the expected profits for Rachel on each case that reflects Emma's effort level. Which
case does Rachel prefer?
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Follow-up Question
b)Rachel’s objective is to maximize the expected profit, subject to that Emma works for Rachel
and Emma puts effort. However, effort level is not observable. Hence, Rachel needs to write
a contract based on the observables. Let’s say, Rachel pays Emma based on the outcome: xL
when the profit is $0, xM when the profit is $2000, and xH when the profit is $3,000. Then
Emma has three options:
(i) Not to work for Rachel
(ii) Work for Rachel without effort
(iii) Work for Rachel with effort
Find Emma’s expected utility on each option
c)ssuming Rachel wants Emma to put effort, her objective essentially becomes to find the
lowest contingent payment scheme that is just enough for Emma to work for Rachel, and
gives an incentive for Emma to put effort. Formally, we can write this as:
min 0.1xL + 0.3xM + 0.6xH ,
subject to
0.1√xL + 0.3√xM + 0.6√xH − 5 ≥ 15, (1)
and
0.1√xL + 0.3√xM + 0.6√xH − 5 ≥ 0.6√xL + 0.3√xM + 0.1√xH . (2)
What is Constraint (1) called? What is Constraint (2) called?
d)
or your information, the solution for (c) is:
xL = $130.60
xM = $400
xH = $459.16.
xL = $130.60
xM = $400
xH = $459.16.
This contract will maximize Rachel’s expected profits.
There is another way to achieve Rachel’s objective. Instead of specifying the payment on
each outcome, Rachel can offer Emma a flat payment with a bonus. For example, Rachel
offers $225. This flat payment is enough for Emma to work for Rachel. However, if this is
the only payment, Emma does not have any incentive to put effort. For a better outcome,
Rachel needs to make sure Emma put her effort. To induce her effort, Rachel can promise
some bonus as some percentage of $B upon a better outcome. Let’s say the expected profit
is $1200 higher with effort than without effort. Then Rachel sets B = 1200 and promise a
bonus of $xB, where 0 < x < 1, upon a better outcome. To induce Emma’s effort, what is
the minimum x that Rachel promises to Emma?
There is another way to achieve Rachel’s objective. Instead of specifying the payment on
each outcome, Rachel can offer Emma a flat payment with a bonus. For example, Rachel
offers $225. This flat payment is enough for Emma to work for Rachel. However, if this is
the only payment, Emma does not have any incentive to put effort. For a better outcome,
Rachel needs to make sure Emma put her effort. To induce her effort, Rachel can promise
some bonus as some percentage of $B upon a better outcome. Let’s say the expected profit
is $1200 higher with effort than without effort. Then Rachel sets B = 1200 and promise a
bonus of $xB, where 0 < x < 1, upon a better outcome. To induce Emma’s effort, what is
the minimum x that Rachel promises to Emma?
Transcribed Image Text:(c) Assuming Rachel wants Emma to put effort, her objective essentially becomes to find the
lowest contingent payment scheme that is just enough for Emma to work for Rachel, and
gives an incentive for Emma to put effort. Formally, we can write this as:
min 0.1xL+0.3xM +0.6TH,
subject to
0.1√TL +0.3√TM +0.6√√xH-5≥ 15,
0.1√TL +0.3√TM +0.6√TH-5≥ 0.6√xL+0.3√xM +0.1√TH.
What is Constraint (1) called? What is Constraint (2) called?
(d) For your information, the solution for (c) is:
and
XL = $130.60
*M = $400
TH = $459.16.
3
(1)
(2)
This contract will maximize Rachel's expected profits.
There is another way to achieve Rachel's objective. Instead of specifying the payment on
each outcome, Rachel can offer Emma a flat payment with a bonus. For example, Rachel
offers $225. This flat payment is enough for Emma to work for Rachel. However, if this is
the only payment, Emma does not have any incentive to put effort. For a better outcome,
Rachel needs to make sure Emma put her effort. To induce her effort, Rachel can promise
some bonus as some percentage of $B upon a better outcome. Let's say the expected profit
is $1200 higher with effort than without effort. Then Rachel sets B = 1200 and promise a
bonus of $xB, where 0 < x < 1, upon a better outcome. To induce Emma's effort, what is
the minimum x that Rachel promises to Emma?
Solution
by Bartleby ExpertFollow-up Question
Transcribed Image Text:Rachel's objective is to maximize the expected profit, subject to that Emma works for Rachel
and Emma puts effort. However, effort level is not observable. Hence, Rachel needs to write
a contract based on the observables. Let's say, Rachel pays Emma based on the outcome:
when the profit is $0, 2 when the profit is $2000, and zu when the profit is $3,000. Then
Emma has three options:
(i) Not to work for Rachel
(ii) Work for Rachel without effort
(iii) Work for Rachel with effort
Find Emma's expected utility on each option.
Solution
by Bartleby ExpertKnowledge Booster
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